Decomposition of backward SLE in the capacity parameterization

TL;DR

This paper proves a decomposition property of backward chordal SLE$_$ for   4, showing how the law of the process can be expressed as an integral over measures with force points, related to the time spent in certain regions.

Contribution

It establishes a backward SLE decomposition in the capacity parametrization for   4, extending previous forward SLE analyses to the backward case.

Findings

Decomposition of backward chordal SLE$_$ into measures with force points.

Radon-Nikodym derivative relates to the time spent in a region.

Extension of SLE parametrization analysis to backward processes.

Abstract

We prove that, for $\kappa\le 4$, backward chordal SLE$_\kappa$ admits backward chordal SLE$_\kappa(-4,-4)$ decomposition for the capacity parametrization. This means that, for any bounded measurable subset $U\subset Q_4:={\mathbb R}_+\times{\mathbb R}_-$, if we integrate the laws of extended backward chordal SLE$_\kappa(-4,-4)$ with different pairs of force points $(x,y)$ against some suitable density function $G(x,y)$ restricted to $U$, then we get a measure, which is absolutely continuous with respect to the law of backward chordal SLE$_\kappa$, and the Radon-Nikodym derivative is a constant depending on $\kappa$ times the capacity time that the generated welding curve $t\mapsto (d_t,c_t)$ spends in $U$, where $d_t>0>c_t$ are the pair of points that are swallowed by the process at time $t$. For the forward SLE curve, a similar analysis has been done for SLE in the natural parametrization ([1] $\kappa \leq 4$, [10] $\kappa <8$), and for the capacity parametrization ([10] $\kappa < \infty$).